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Properties

Label 2-525-15.2-c1-0-21
Degree $2$
Conductor $525$
Sign $0.461 + 0.886i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.723 − 0.723i)2-s + (−0.994 − 1.41i)3-s + 0.951i·4-s + (−1.74 − 0.307i)6-s + (0.707 + 0.707i)7-s + (2.13 + 2.13i)8-s + (−1.02 + 2.81i)9-s − 3.63i·11-s + (1.34 − 0.946i)12-s + (4.19 − 4.19i)13-s + 1.02·14-s + 1.19·16-s + (4.61 − 4.61i)17-s + (1.30 + 2.78i)18-s + 3.77i·19-s + ⋯
L(s)  = 1  + (0.511 − 0.511i)2-s + (−0.573 − 0.818i)3-s + 0.475i·4-s + (−0.713 − 0.125i)6-s + (0.267 + 0.267i)7-s + (0.755 + 0.755i)8-s + (−0.341 + 0.939i)9-s − 1.09i·11-s + (0.389 − 0.273i)12-s + (1.16 − 1.16i)13-s + 0.273·14-s + 0.297·16-s + (1.11 − 1.11i)17-s + (0.306 + 0.655i)18-s + 0.865i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.461 + 0.886i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.461 + 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47204 - 0.893109i\)
\(L(\frac12)\) \(\approx\) \(1.47204 - 0.893109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.994 + 1.41i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-0.723 + 0.723i)T - 2iT^{2} \)
11 \( 1 + 3.63iT - 11T^{2} \)
13 \( 1 + (-4.19 + 4.19i)T - 13iT^{2} \)
17 \( 1 + (-4.61 + 4.61i)T - 17iT^{2} \)
19 \( 1 - 3.77iT - 19T^{2} \)
23 \( 1 + (-1.81 - 1.81i)T + 23iT^{2} \)
29 \( 1 + 1.13T + 29T^{2} \)
31 \( 1 - 3.62T + 31T^{2} \)
37 \( 1 + (7.24 + 7.24i)T + 37iT^{2} \)
41 \( 1 - 0.314iT - 41T^{2} \)
43 \( 1 + (-1.06 + 1.06i)T - 43iT^{2} \)
47 \( 1 + (4.48 - 4.48i)T - 47iT^{2} \)
53 \( 1 + (1.44 + 1.44i)T + 53iT^{2} \)
59 \( 1 - 8.70T + 59T^{2} \)
61 \( 1 - 3.08T + 61T^{2} \)
67 \( 1 + (-9.67 - 9.67i)T + 67iT^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (0.710 - 0.710i)T - 73iT^{2} \)
79 \( 1 - 7.30iT - 79T^{2} \)
83 \( 1 + (9.58 + 9.58i)T + 83iT^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + (5.28 + 5.28i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.18381346213784985652750161834, −10.19358701915624246394576902637, −8.540912582596191161716297281951, −8.070158393859081395213366559169, −7.14040162689977602619721571187, −5.73124129397974750806718909391, −5.32787810060182895506480065604, −3.67186692430915324172601396109, −2.77167543675447156978712869942, −1.16697801777032980557440145382, 1.44366194123726876279549146397, 3.72078298890790141359335074866, 4.52903752168551481812560164147, 5.29377136972952513924434154202, 6.38530796521701476179352294443, 6.92382603482248239916699777492, 8.386525167298518513023666525559, 9.474479728343579657341903128940, 10.19462070282807430148807612330, 10.89139444942269976059537883180

Graph of the $Z$-function along the critical line